In a paper of 't Hooft about the rôle of magnetic monopoles for a model of quark confinement, I don't understand the following sentence (end at paragraph 14)

[...] in order for monopoles to produce a magnetic Higgs mechanism they must become very light.

What is to background for this statement? Why do they need to be (very) light?


I do think that it is related to the ype of superconductor you need to have a confining effect. For field generated by the pairs of quarks to be confined, you need flux tubes to be allowed inside condensed monopole-medium, i.e. you need a type II superconductor.

From Wikipedia I read that the type of SC is determined by:

The ratio of the London penetration depth λ to the superconducting coherence length ξ determines whether a superconductor is type-I or type-II. Type-I superconductors are those with 0 < λ/ξ < 1/√2, and type-II superconductors are those with λ/ξ > 1/√2

Although it is not explicitly shown in the article I suppose that the attenuation length must be inversely proportional to the mass of the monopoles of the theory...

Is this right? (If this is the case could someone give me a reference?)


G. t' Hooft, The Topological Mechanism for Permanent Quark Confinement in a Non-Abelian Gauge Theory. (Link to the article )

  • $\begingroup$ I'm not sure I can help you so much, but the vortex state is well described in the book by Tinkham (Superconductivity). $\endgroup$ – FraSchelle May 21 '14 at 16:21

This is fully contained in the paper, from understanding the sections that came before your statement. Namely:

Section 5 of the paper studies QED using "perturbative expansion" which requires the charge $e$ of the electron to be small, and this is how the Higgs mechanism (with respect to electrons) occurs.

From Dirac's quantization condition, the charge $q$ (and hence mass) of the monopole must then be large. Now Section 9+10 clarifies how we can look at this system "dually" (the symmetry present in Maxwell's equations), for which $q$ is small and $e$ is large. So a "formal repeat" of Section 5 will give you the "magnetic Higgs mechanism" when monopole mass is small. Sections 11 to 14 work out those formalities.

tl;dr The "magnetic Higgs mechanism" pops up from doing a "perturbative expansion" of QED with respect to magnetic monopoles, and that only makes sense if the mass/charge is very small.

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